AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Stage One: Exploring and Developing Interesting Results
That Lead to Conjectures
Introduction
Goals of Stage One
• Estimate distance traveled from a chart of positive velocity values
• Investigate properties of integral functions when the integrand is a
positive constant
Articles in Stage One
• “Key Ideas That Help Students Understand the Fundamental Theorem”
by Steve Olson
• “Functions Defined by an Integral” by Mark Howell
Keeping the context simple is an important first principle. When the context is simple,
students can see and easily recognize the new and important concepts. Many teachers
have found that velocity examples are particularly good for introducing the idea of the
definite integral. Students are familiar with and have had practical experiences using
velocity. The formula, distance = rate × time, is easy to remember and apply. At this stage,
we use this formula to estimate the distance traveled based on a table of velocity values.
We begin with positive velocity values in the first chapter, by Steve Olson. This article
presents specific examples and gives some tips that will encourage students to explore
and experiment with velocity to make conjectures about distance and displacement.
Mark Howell’s “Functions Defined by a Definite Integral” considers functions of the
x
form f ( x)= ∫ k dt when k is a positive constant, a is any constant, and x ≥ a. The use
a
of a positive constant k parallels the positive velocity values in the first article. Students
will experiment with constant integrands and several lower limits to discover the
relationship between the integrands and the slope of the linear functions defined by
the definite integrals.
These two settings, positive velocity and integrals with constant integrands, are comfortable
ones for students, and that makes it possible to informally introduce deep concepts.
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AP® Calculus: 2006–2007 Workshop Materials